Previous Colloquia - London Heilbronn Colloquiua. Online registration by Cvent

Since the twin breakthroughs in 1982-83 by Freedman and Donaldson the study of four manifolds has been developing rapidly. Freedman’s resolution of the 4d topological Poincaré conjecture and refinements combined with Donaldson’s surprising applications of the Yang-Mills equations to show that the situation for smooth structures in dimension 4 was more complicated than in higher dimensions. Since then new theories including Seiberg-Witten equations, Ozsvath and Szabó’s Heegaard Floer theory, and Embedded Contact Homology and have led to surprising applications to questions in 3 and 4 dimensional topology. This talk will survey some of these developments.

Tomasz Mrowka’s research interests focus on problems in differential geometry and gauge theory. His work combines analysis, geometry, and topology, specializing in the use of partial differential equations, such as the Yang-Mills equations from particle physics to analyze low-dimensional mathematical objects. Jointly with Robert Gompf, he discovered four-dimensional models of space-time topology.

A graduate of MIT, Mrowka received the Ph.D. from U.C. Berkeley in 1988 under the direction of Clifford Taubes and Robin Kirby. He joined the MIT mathematics faculty as professor in 1996, following faculty appointments at Stanford and at Caltech (professor 1994-96). He chaired the Graduate Student Committee 1999-02, and chaired the Pure Mathematics Committee, 2004-15. From 2014-2017 he served as Department Head. A prior Sloan fellow and Young Presidential Investigator, Mrowka was selected for a Clay Mathematics Visiting Professorship in 1995.

In 2007 he received the Oswald Veblen Prize in Geometry by the AMS, jointly with Peter Kronheimer, “for their joint contributions to both three- and four- dimensional topology through the development of deep analytical techniques and applications.” Their book, Monopoles and Three Manifolds (Cambridge University Press) also garnered the 2011 Joseph Doob Prize of the AMS. He was appointed Singer Professor of Mathematics from 2007 to 2017. In 2017, Mrowka received a Simons Fellowship in Mathematics. In 2018 delivered a plenary address at ICM18 in Rio de Janeiro. He is a Fellow of the American Academy of Arts & Sciences (2007) and Member of the National Academy of Sciences (2015). Most recently, he was awarded the 2023 Leroy P. Steele Prize for Seminal Contribution to Research for his joint paper with Peter Kronheimer, ‘Gauge theory for embedded surfaces, I’ published in 1993 in Topology.

Are there infintely many quadratic characters (for instance, the Legendre symbol mod p) for which the partial sums are always non-negative? Although only 0% of characters can have this property, numerical work (most recently by Kalmynin) suggests that such characters are nevertheless plentiful. For instance, computations show that there are many more examples than may be expected by modeling such sums by a simple random walk. I will discuss joint work with Angelo and Xu which obtains new upper bounds for the number of such characters, by studying a related problem on maxima of L-values (closely connected to the Fyodorov-Hiary-Keating conjectures). I will give a heuristic explanation for why such characters are plentiful, and which suggests that our upper bounds may not be too far from the truth.

Many problems in number theory can be framed as questions about counting integral solutions to a Diophantine equation (say, within a certain “box”). If there are very few, or very many variables, certain well-established methods gain an advantage. But sometimes there is extra structure that can be exploited as well. For example: let f be a given polynomial with integer coefficients in n variables. How many values of f are a perfect square? A perfect cube? Or, more generally, a value of a different polynomial of interest, say g? These questions arise in a variety of specific applications, and also in the context of a general conjecture of Serre on counting points in thin sets. We will describe how sieve methods can exploit this type of structure, and explain how a new polynomial sieve method allows greater flexibility, so that the variables in the polynomials f and g can “mix.”

This is joint work with Dante Bonolis.

Lillian Pierce received a BA in Mathematics and graduated as valedictorian of Princeton University in 2002. She earned an MSc by Research at the University of Oxford as a Rhodes Scholar in 2004, and a PhD from Princeton in 2009. After postdoctoral positions at the Institute for Advanced Study and University of Oxford and a year as a Bonn Junior Fellow, Pierce took a faculty position at Duke University, where she is presently a Professor of Mathematics. Pierce’s research combines techniques of analytic number theory and harmonic analysis, with particular interests in Diophantine equations, exponential and character sums, class groups, oscillatory integrals, and singular integrals. Pierce has recently founded the journal Essential Number Theory, which aims to deepen the impact of important ideas by encouraging authors to write clear, useful expositions. Pierce’s work has been recognized by a Presidential Early Career Award for Scientists and Engineers, a Simons Fellowship, a Joan and Joseph Birman Fellowship, a Sloan Research Fellowship, a von Neumann Fellowship, and a Marie Curie Fellowship. She was an invited speaker, representing number theory and analysis, at the International Congress of Mathematicians in 2022. Her work in 2024 will be supported by a Simons Research Fellowship and Guggenheim Fellowship.

Two of the most striking discoveries of 18th and 19th century number theory are the Kronecker-Weber theorem and the theory of complex multiplication. The first asserts that the maximal abelian extension of the field Q of rational numbers is generated by roots of unity – in other words, that all abelian extensions of Q can be constructed by adjoining values at rational arguments of the transcendental function

e(z) := e^{2pi iz} = cos(2 pi z) + i sin(2 pi z).

The second achieves something similar for a quadratic imaginary field K, constructing essentially all of its abelian extensions from values of the modular j-function at arguments of K.

Finding analytic functions that would play the role of trigonometric and modular functions in generating abelian extensions, or class fields, of more general base fields is the somewhat loosely formulated program of explicit class field theory, also known as Kronecker’s Jugendtraum or Hilbert’s twelfth problem.

Partial progress was achieved with the theory of complex multiplication of abelian varieties initiated by Hilbert and his school and brought to maturity in the eponymous 1961 treatise of Shimura and Taniyama. Through this theory, class fields of CM fields are obtained from the values of modular functions at points attached to the moduli of CM abelian varieties in suitable (Hilbert, Siegel, orthogonal, . . .) Shimura varieties.

Hilbert’s twelfth problem for non-CM base fields remains shrouded in a great deal of mystery, hinting—perhaps—at a rich function theory for arithmetic quotients even when the underlying real symmetric space fails to be endowed with a complex structure and hence cannot uniformise a Shimura variety. This may be what Hilbert intuited when he declared, in his celebrated 1900 address at the Paris ICM,

“I am certain that the theory of analytical functions of several variables in particular would be notably enriched if one should succeed in finding and discussing those functions which play the part for any algebraic number field corresponding to that of the exponential function in the field of rational numbers and of the elliptic modular functions in the imaginary quadratic number field.”

This talk will describe some possible substitutes for trigonometric and modular functions, with applications to explicit class field theory beyond the traditional framework of CM fields.

General relativity makes spectacular predictions about our world, predictions which have captured the popular imagination more than any other part of physics: gravitational waves, black holes, spacetime singularities. For the mathematician, however, perhaps the most spectacular thing about these predictions is not their exoticness, but, on the contrary, the fact that they all correspond to well-defined mathematical concepts: Indeed, it was precisely through mathematics that these predictions of general relativity were first discovered—originally to much controversy and objection!—and the qualitative mathematical analysis of the Einstein equations remains one of the most powerful ways to understand the great conceptual questions of the theory. This talk will describe some past contributions of mathematics to general relativity and some of the big open conjectures which mathematics hopes to answer in the future.

The Shimura-Taniyama-Weil modularity conjecture asserts that all elliptic curves over Q arise as images of quotients of the Poincare upper half plane by congruence subgroups of the modular group SL₂(Z). Wiles proved Fermat's Last Theorem by establishing the modularity of semistable elliptic curves over Q. Subsequent work of Breuil-Conrad-Diamond-Taylor established the modularity of elliptic curves over Q in full generality. My work with J-P. Wintenberger gave a proof of the generalized Shimura-Taniyama-Weil conjecture which asserts that all "odd, rank 2 motives over Q" are modular. This is a corollary of our proof of Serre's modularity conjecture.

Very little is known when one looks at the same question over finite extensions of Q. I will talk about the recent beautiful work of Ana Caraiani and James Newton which proves modularity of all elliptic curves over Q(i). An input into their proof is a result, proved in joint work with Patrick Allen and Jack Thorne, that proves the analog of Serre's conjecture for mod 3 representations that arise from elliptic curves over Q(i).

My talk will give a general introduction to this circle of ideas centred around the modularity conjecture for motives and Galois representations over number fields. We know only fragments of what is conjectured, but what little we know is already quite remarkable!

Chandrashekhar Khare was born in Mumbai, and studied at Cambridge, Oxford and Caltech, where he obtained his Ph.D. In 1995. He worked at the Tata Institute of Fundamental Research and the University of Utah, and is now a professor at the University of California at Los Angeles. His research is in number theory, especially on the relation between modular forms and Galois representations that underpins Wiles’ proof of Fermat’s Last Theorem; in 2008, he and Jean-Pierre Winterberger made a remarkable breakthrough with their proof of a celebrated conjecture of J.-P. Serre. Prof. Khare’s honours and awards include the Fermat Prize (2007), Infosys Prize (2010) and the Cole Prize (2011), and he was elected as a Fellow of the Royal Society in 2012.